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Register a new userCommon kernel-smoothed proper orthogonal decomposition (CKSPOD): An efficient reduced-order model for emulation of spatiotemporally evolving flow dynamics
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In the present study, we propose a new surrogate model, called common kernel-smoothed proper orthogonal decomposition (CKSPOD), to efficiently emulate the spatiotemporal evolution of fluid flow dynamics. The proposed surrogate model integrates and extends recent developments in Gaussian process learning, high-fidelity simulations, projection-based model reduction, uncertainty quantification, and experimental design, rendering a systematic, multidisciplinary framework. The novelty of the CKSPOD emulation lies in the construction of a common Gram matrix, which results from the Hadamard product of Gram matrices of all observed design settings. The Gram matrix is a spatially averaged temporal correlation matrix and contains the temporal dynamics of the corresponding sampling point. The common Gram matrix synthesizes the temporal dynamics by transferring POD modes into spatial functions at each observed design setting, which remedies the phase-difference issue encountered in the kernel-smoothed POD (KSPOD) emulation, a recent fluid flow emulator proposed in Chang et al. (2020). The CKSPOD methodology is demonstrated through a model study of flow dynamics of swirl injectors with three design parameters. A total of 30 training design settings and 8 validation design settings are included. Both qualitative and quantitative results show that the CKSPOD emulation outperforms the KSPOD emulation for all validation cases, and is capable of capturing small-scale wave structures on the liquid-film surface faithfully. The turbulent kinetic energy prediction using CKSPOD reveals lower predictive uncertainty than KSPOD, thereby allowing for more accurate and precise flow predictions. The turnaround time of the CKSPOD emulation is about 5 orders of magnitude faster than the corresponding high-fidelity simulation, which enables an efficient and scalable framework for design exploration and optimization.
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This interdisciplinary study, which combines machine learning, statistical methodologies, high-fidelity simulations, and flow physics, demonstrates a new process for building an efficient surrogate model for predicting spatiotemporally evolving flow dynamics. In our previous work, a common-grid proper-orthogonal-decomposition (CPOD) technique was developed to establish a physics-based surrogate (emulation) model for prediction of mean flowfields and design exploration over a wide parameter space. The CPOD technique is substantially improved upon here using a kernel-smoothed POD (KSPOD) technique, which leverages kriging-based weighted functions from the design matrix. The resultant emulation model is then trained using a dataset obtained through high-fidelity simulations. As an example, the flow evolution in a swirl injector is considered for a wide range of design parameters and operating conditions. The KSPOD-based emulation model performs well, and can faithfully capture the spatiotemporal flow dynamics. The model enables effective design surveys utilizing high-fidelity simulation data, achieving a turnaround time for evaluating new design points that is 42,000 times faster than the original simulation.
An extension of Proper Orthogonal Decomposition is applied to the wall layer of a turbulent channel flow (Re {tau} = 590), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the igenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200-300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12 u {tau} in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects.
The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a 2D wake flow and a 3D flow generated by a flapping insect computed with direct numerical simulation.
We investigate the applicability of machine learning based reduced order model (ML-ROM) to three-dimensional complex flows. As an example, we consider a turbulent channel flow at the friction Reynolds number of $Re_tau=110$ in a minimum domain which can maintain coherent structures of turbulence. Training data set are prepared by direct numerical simulation (DNS). The present ML-ROM is constructed by combining a three-dimensional convolutional neural network autoencoder (CNN-AE) and a long short-term memory (LSTM). The CNN-AE works to map high-dimensional flow fields into a low-dimensional latent space. The LSTM is then utilized to predict a temporal evolution of the latent vectors obtained by the CNN-AE. The combination of CNN-AE and LSTM can represent the spatio-temporal high-dimensional dynamics of flow fields by only integrating the temporal evolution of the low-dimensional latent dynamics. The turbulent flow fields reproduced by the present ML-ROM show statistical agreement with the reference DNS data in time-ensemble sense, which can also be found through an orbit-based analysis. Influences of the population of vortical structures contained in the domain and the time interval used for temporal prediction on the ML- ROM performance are also investigated. The potential and limitation of the present ML-ROM for turbulence analysis are discussed at the end of our presentation.
In recent years, there have been a surge in applications of neural networks (NNs) in physical sciences. Although various algorithmic advances have been proposed, there are, thus far, limited number of studies that assess the interpretability of neural networks. This has contributed to the hasty characterization of most NN methods as black boxes and hindering wider acceptance of more powerful machine learning algorithms for physics. In an effort to address such issues in fluid flow modeling, we use a probabilistic neural network (PNN) that provide confidence intervals for its predictions in a computationally effective manner. The model is first assessed considering the estimation of proper orthogonal decomposition (POD) coefficients from local sensor measurements of solution of the shallow water equation. We find that the present model outperforms a well-known linear method with regard to estimation. This model is then applied to the estimation of the temporal evolution of POD coefficients with considering the wake of a NACA0012 airfoil with a Gurney flap and the NOAA sea surface temperature. The present model can accurately estimate the POD coefficients over time in addition to providing confidence intervals thereby quantifying the uncertainty in the output given a particular training data set.
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